Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.192.1.774 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&36\\32&37\end{bmatrix}$, $\begin{bmatrix}33&36\\24&27\end{bmatrix}$, $\begin{bmatrix}39&12\\22&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.96.1.bc.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $3840$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 y^{2} + 2 y z + 2 z^{2} + w^{2} $ |
| $=$ | $5 x^{2} - y^{2} + y z + z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 4 x^{2} y^{2} + 15 x^{2} z^{2} + 9 y^{4} + 30 y^{2} z^{2} + 25 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^4}{3^8\cdot5^2}\cdot\frac{147224000000000000yz^{23}+485839200000000000yz^{21}w^{2}+695633400000000000yz^{19}w^{4}+566444340000000000yz^{17}w^{6}+286709673600000000yz^{15}w^{8}+91217145600000000yz^{13}w^{10}+16933693140000000yz^{11}w^{12}+1261308510000000yz^{9}w^{14}-112930031520000yz^{7}w^{16}-23172402240000yz^{5}w^{18}-51325390800yz^{3}w^{20}+79099678440yzw^{22}+188141000000000000z^{24}+662585200000000000z^{22}w^{2}+1020383910000000000z^{20}w^{4}+902225070000000000z^{18}w^{6}+501489873000000000z^{16}w^{8}+177358118400000000z^{14}w^{10}+36898968015000000z^{12}w^{12}+2905981353000000z^{10}w^{14}-459169354260000z^{8}w^{16}-106968444480000z^{6}w^{18}+125736972300z^{4}w^{20}+1085135206140z^{2}w^{22}-40209003207w^{24}}{w^{8}(316000000yz^{15}+663600000yz^{13}w^{2}+554580000yz^{11}w^{4}+234630000yz^{9}w^{6}+53119800yz^{7}w^{8}+6342300yz^{5}w^{10}+376164yz^{3}w^{12}+8748yzw^{14}+598750000z^{16}+1405400000z^{14}w^{2}+1339455000z^{12}w^{4}+664389000z^{10}w^{6}+183649275z^{8}w^{8}+28588950z^{6}w^{10}+2499741z^{4}w^{12}+115182z^{2}w^{14}+2187w^{16})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.96.1.bc.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+4X^{2}Y^{2}+9Y^{4}+15X^{2}Z^{2}+30Y^{2}Z^{2}+25Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.m.2.8 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.0-40.i.1.8 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.i.1.15 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.k.2.3 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.k.2.10 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.q.2.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.q.2.12 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.s.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.s.1.16 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.1-8.m.2.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.u.1.5 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.u.1.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.w.1.14 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.w.1.16 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.960.33-40.dc.2.3 | $40$ | $5$ | $5$ | $33$ | $7$ | $1^{14}\cdot2^{9}$ |
| 40.1152.33-40.kg.1.3 | $40$ | $6$ | $6$ | $33$ | $3$ | $1^{14}\cdot2\cdot4^{4}$ |
| 40.1920.65-40.oc.1.1 | $40$ | $10$ | $10$ | $65$ | $11$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |